The classical solution of arbitrary smoothness for the first mixed problem for the Klein-Gordon-Fock type equation. (Russian. English summary) Zbl 1532.35292
Summary: In this paper, we consider the first mixed problem for the one-dimensional Klein-Gordon-Fock type equation in a half-strip. Meanwhile, the existence and uniqueness of a solution of arbitrary smoothness is researched. While solving this problem using the method of characteristics, equivalent second type Volterra integral equations appear. The existence of a unique solution in the class of \(n\) times continuously differentiable functions is proven for these equations when initial functions are smooth enough. Moreover, it is shown that for the smoothness of the solution of the initial problem it is necessary and sufficient that the matching conditions for the given functions be fulfilled if they are sufficiently smooth. The method of characteristics, used for problem analysis, is reduced to separating the total area of the solution on subdomains in each of them so that the solution of the subproblem is constructed with the help of the initial and boundary conditions. Then, the obtained solutions are glued in common points, and the received glued conditions are the matching conditions. This approach permits to construct both exact and approximate solutions. The exact solutions can be found when it is possible to solve the equivalent Volterra integral equations. Otherwise, one can find an approximate solution of the problem either in analytical or numerical form. Along with this, when constructing an approximate solution, the matching conditions turn out to be essential, which must be taken into account when using numerical methods for solving the problem.
MSC:
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35C05 | Solutions to PDEs in closed form |
Keywords:
Klein-Gordon-Fock equation; method of characteristics; classical solution; first mixed problem; matching conditions; one space dimensionReferences:
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